On the Number of Synchronizing Colorings of Digraphs

TL;DR

This paper investigates the number of synchronizing colorings in $k$-out-regular digraphs, combining experimental enumeration with theoretical conjectures, and explores the fraction of such colorings in various digraphs.

Contribution

It provides the first extensive experimental analysis of synchronizing colorings in small digraphs and proposes new conjectures about their minimal fractions.

Findings

Identified digraphs with a fraction of synchronizing colorings approaching 1-1/k.

Constructed digraphs with fractions exactly equal to 1-1/k^d.

Formulated conjectures about the minimal possible fraction of synchronizing colorings.

Abstract

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.